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An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth

Marc Distel

TL;DR

This work advances the coarse-geometry understanding of graphs quasi-isometric to bounded-treewidth graphs by giving a structural characterization via partitions whose quotient has bounded treewidth and whose parts have bounded weak diameter. Central to the approach is the notion of an $\ell$-compressing $H$-indexed partition and a compressing function for the class of graphs with treewidth $\le k$ (proved to be $\ell \mapsto 2(k+1)\ell$). The authors construct a hierarchical center-based partition along a tree-decomposition, culminating in a quotient $H$ with width at most $k$ and parts of weak diameter at most $2(k+1)\ell$, which yields a broad, complementary characterization to prior results. These insights enrich the toolbox for analyzing quasi-isometries and have potential implications for understanding the large-scale structure of sparse graphs. All mathematical notation is kept within $...$ delimiters.

Abstract

Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of treewidth $k$. In this paper, we give a new characterisation of such graphs. Specifically, we show that such graphs $G$ are characterised by the existence of a partition whose quotient has treewidth at most $k$ and such that each part has bounded weak diameter in $G$. The primary contribution of our characterisation is a structural description of graphs that admit such a quasi-isometry. This differs from the characterisation mentioned above, which primarily shows the existence of such a quasi-isometry. The characterisations are complementary, and neither immediately implies the other.

An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth

TL;DR

This work advances the coarse-geometry understanding of graphs quasi-isometric to bounded-treewidth graphs by giving a structural characterization via partitions whose quotient has bounded treewidth and whose parts have bounded weak diameter. Central to the approach is the notion of an -compressing -indexed partition and a compressing function for the class of graphs with treewidth (proved to be ). The authors construct a hierarchical center-based partition along a tree-decomposition, culminating in a quotient with width at most and parts of weak diameter at most , which yields a broad, complementary characterization to prior results. These insights enrich the toolbox for analyzing quasi-isometries and have potential implications for understanding the large-scale structure of sparse graphs. All mathematical notation is kept within delimiters.

Abstract

Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of treewidth . In this paper, we give a new characterisation of such graphs. Specifically, we show that such graphs are characterised by the existence of a partition whose quotient has treewidth at most and such that each part has bounded weak diameter in . The primary contribution of our characterisation is a structural description of graphs that admit such a quasi-isometry. This differs from the characterisation mentioned above, which primarily shows the existence of such a quasi-isometry. The characterisations are complementary, and neither immediately implies the other.
Paper Structure (6 sections, 9 theorems, 7 equations)

This paper contains 6 sections, 9 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compressing Partitions
  4. Proof Explanation
  5. Proof
  6. Acknowledgements.

Key Result

Theorem 1

Let $k\in \mathbb{N}$ and $\ell\in \mathbb{R}^{\geqslant0}$. If a graph $G$ admits a tree-decomposition where each bag is the union of at most $k$ sets each of weak diameter at most $\ell$, then $G$ is $2(k+2)\ell$-quasi-isometric to a graph of treewidth at most $k$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 18 more